How can I train a neural network for another input set, without losing the learning of the previous input set?
Yes, this is actually a limitation known as catastrophic forgetting. 
A proposed way to deal with this is elastic weight consolidation that "remembers old tasks by selectively slowing down learning on the weights important for those tasks". See Overcoming catastrophic forgetting in neural networks for details. Another approach is Learning without forgetting.

If the tasks are different, the approach you are talking about is called transfer learning. You might want to have a look at multi-task learning as well 

If the tasks are the same, you could try creating a join of both datasets and training on that.